Octahedron - Area and Volume

Area and Volume

The surface area A and the volume V of a regular octahedron of edge length a are:

Thus the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice (because we have 8 vs. 4 triangles).

If an octahedron has been stretched so that it obeys the equation:

The formula for the surface area and volume expand to become:

Additionally the inertia tensor of the stretched octahedron is:

I = \begin{bmatrix} \frac{1}{10} m (y_m^2+z_m^2) & 0 & 0 \\ 0 & \frac{1}{10} m (x_m^2+z_m^2) & 0 \\ 0 & 0 & \frac{1}{10} m (x_m^2+y_m^2) \end{bmatrix}

These reduce to the equations for the regular octahedron when:

Read more about this topic:  Octahedron

Famous quotes containing the words area and/or volume:

    “The area [of toilet training] is one where a child really does possess the power to defy. Strong pressure leads to a powerful struggle. The issue then is not toilet training but who holds the reins—mother or child? And the child has most of the ammunition!”
    —Dorothy Corkville Briggs (20th century)

    “Bishop Berkeley destroyed this world in one volume octavo; and nothing remained, after his time, but mind; which experienced a similar fate from the hand of Hume in 1737.”
    —Sydney Smith (1771–1845)